Calibrated Intrusive Reduced-Order Model of Burgers’ Equation Using a Combination of Proper Orthogonal Decomposition and LSTM Deep Learning Algorithm

Modelling plays a critical role in many engineering applications. Partial differential equations (PDEs) are ubiquitous, describing various physical phenomena such as fluid flow, electromagnetism, and quantum mechanics. Although some of these equations have analytical solutions, many require high-fidelity simulations of parametric PDEs. In general, high-fidelity simulations are computationally expensive and often infeasible for real-time or multi-query applications. This challenge has led to the development of reduced-order models (ROMs). Over the past few decades, ROMs have emerged as a practical solution for simulating, controlling, and optimizing large-scale and complex dynamical systems. This paper introduces a novel Calibrated Intrusive Reduced-Order Modelling (CIROM) approach for the efficient and accurate simulation of the one-dimensional Burgers’ equation, employed as a canonical benchmark because it is a simplified fundamental partial differential equation that captures the behaviour of many real-world phenomena. The proposed method, combining the strengths of proper orthogonal decomposition (POD) and long short-term memory (LSTM) networks, effectively reduces computational complexity while addressing inherent instabilities in classical reduced-order models. Unlike traditional POD-ROMs, which often suffer from error accumulation and instability at high Reynolds numbers, the CIROM employs an iterative LSTM-based error correction mechanism to learn and compensate for truncation and projection errors. This study is benchmark-oriented and does not aim to provide a general PDE solver. The performance of the proposed method is rigorously evaluated across a broad range of Reynolds numbers, including interpolation and extrapolation scenarios, demonstrating robust extrapolation within moderate ranges. Comprehensive numerical experiments confirm that the CIROM outperforms both pure intrusive ROMs and purely data-driven LSTM models in terms of prediction accuracy, stability, and computational cost.